Derivatives of matrix order (Q1369324)

The classical theory of fractional calculus deals with the investigation and applications of derivatives and integrals of arbitrary real or complex order -- see classical and modern results and bibliography in the book by \textit{S. G. Samko}, \textit{A. A. Kilbas} and \textit{O. I. Marichev}: ``Fractional integrals and derivatives. Theory and applications'' (Russian, 1987; Zbl 0617.26004; English translation, 1993; Zbl 0818.26003). The present paper is the attempt to develop an analogy of the fractional calculus in the framework of the matrix calculus. Using the decomposition of the square semisimple matrix \(M\) in the form \(M= \sum_j(\lambda_jP_j+ N_j)\), where \(\lambda_j\) are the eigenvalues and \(P_j\) and \(N_j\) are the corresponding projection and nilpotent matrices, the authors define the derivative of matrix order \(D^M\) as \(D^M= \sum_j P_jD^{\lambda_j}(e^{N_j\ln D})\). Examples are given and the matrix analogy of the differential equation for the Legendre function is constructed.